Sequences

A sequence is an ordered list of numbers where each number is called a term of the sequence.

Definition

A sequence is a function whose domain is the set of positive integers (or non-negative integers). It is typically denoted as:

$${a_k}_{k=1}^{\infty} = a_1, a_2, a_3, \ldots, a_k, \ldots$$

Where $a_k$ is the $k$th term (or general term) of the sequence.

Types of Sequences

Convergent Sequences

[!success] Definition A sequence ${a_k}$ is convergent if: $$\lim_{k \to \infty} a_k = L$$ where $L$ is a finite real number.

The sequence converges to L as $k$ approaches infinity.

Examples of Convergent Sequences

Sequence General Term Limit Behavior
$1.1, 1.01, 1.001, \ldots$ $a_k = 1 + (0.1)^k$ $1$ Converges to 1
$1, \frac{1}{2}, \frac{1}{3}, \ldots$ $a_k = \frac{1}{k}$ $0$ Converges to 0
$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots$ $a_k = \frac{k}{k+1}$ $1$ Converges to 1

Divergent Sequences

[!warning] Definition A sequence ${a_k}$ is divergent if: $$\lim_{k \to \infty} a_k \text{ does not exist}$$

This includes sequences that:

  • Grow without bound ($\to \infty$ or $\to -\infty$)
  • Oscillate without approaching any value

Examples of Divergent Sequences

Sequence General Term Limit Behavior
$1, 3, 5, 7, \ldots$ $a_k = 2k - 1$ $\infty$ Diverges to $+\infty$
$2, 4, 8, 16, \ldots$ $a_k = 2^k$ $\infty$ Diverges (exponential growth)
$-1, 1, -1, 1, \ldots$ $a_k = (-1)^k$ DNE Diverges (oscillates)

Convergence Tests for Sequences

Basic Limit Approach

The primary method to determine convergence:

  1. Identify the general term $a_k$
  2. Evaluate $\lim_{k \to \infty} a_k$
  3. If the limit exists and is finite → convergent
  4. If the limit is infinite or doesn't exist → divergent

Useful Limits

$$\lim_{k \to \infty} \frac{1}{k^p} = 0 \quad \text{for } p > 0$$

$$\lim_{k \to \infty} r^k = 0 \quad \text{for } |r| < 1$$

$$\lim_{k \to \infty} \left(1 + \frac{1}{k}\right)^k = e$$

Common Sequence Types

Arithmetic Sequences

$$a_k = a_1 + (k-1)d$$

where $d$ is the common difference.

Note: Arithmetic sequences with $d \neq 0$ are always divergent.

Geometric Sequences

$$a_k = a_1 \cdot r^{k-1}$$

where $r$ is the common ratio.

Condition Behavior
$|r| < 1$ Converges to 0
$r = 1$ Converges to $a_1$
$r = -1$ Diverges (oscillates)
$|r| > 1$ Diverges (unbounded)

Related Concepts